Determinant formulas
Master Determinant through 28 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.
Determinant, every formula
28 formulas, typeset and free. Print it, or keep it open beside your practice.
Determinant 2×2
Q1Numerical2×2The value of $\begin{vmatrix}3&2\\1&4\end{vmatrix}$ is:Determinant 3×3 (row expansion)
Q1MCQRow expansionA $3\times3$ determinant expanded along row 1 is:- A$a_{11}A_{11}+a_{12}A_{12}+a_{13}A_{13}$
- B$a_{11}+a_{12}+a_{13}$
- C$a_{11}M_{11}+a_{12}M_{12}+a_{13}M_{13}$
- D$A_{11}A_{12}A_{13}$
- A
Minor
Q1MCQMinorThe minor $M_{ij}$ is the determinant obtained by:- Adeleting row $i$ and column $j$
- Bdeleting column $j$ only
- Ctransposing
- Dmultiplying row $i$
- A
Cofactor
Q1NumericalCofactor signThe sign factor $(-1)^{i+j}$ for the $(1,2)$ cofactor is (enter $+1$ or $-1$):Transpose invariance
Q1MCQTranspose$|A^{T}|$ equals:- A$|A|$
- B$-|A|$
- C$\dfrac{1}{|A|}$
- D$|A|^{2}$
- A
Row/column swap
Q1MCQRow swapSwapping two rows of a determinant:- Achanges its sign
- Bleaves it unchanged
- Cmakes it zero
- Ddoubles it
- A
Two identical rows
Q1NumericalIdentical rowsThe value of a determinant with two identical rows is:Scalar multiple of a row
Q1MCQScalar rowIf one row of $|A|$ is multiplied by $5$, the determinant becomes:- A$5|A|$
- B$|A|$
- C$25|A|$
- D$|A|+5$
- A
Determinant of kA
n = order
Q1Numerical|kA|For a $3\times3$ matrix with $|A|=2$, $|3A|$ is:Invariant row operation
Q1MCQInvariant operationThe operation $R_1\to R_1+2R_2$ on a determinant:- Aleaves it unchanged
- Bdoubles it
- Cchanges its sign
- Dmakes it zero
- A
Determinant of a product
Q1Numerical|AB|If $|A|=3$ and $|B|=4$, then $|AB|$ is:Triangular determinant
Q1NumericalTriangularThe value of $\begin{vmatrix}2&5\\0&3\end{vmatrix}$ is:Area of a triangle
Q1NumericalAreaThe area of the triangle with vertices $(0,0),(4,0),(0,6)$ (via determinant) is:Collinearity condition
Q1MCQCollinearityThree points are collinear iff the $3\times3$ coordinate determinant equals:- A$0$
- B$1$
- C$\neq0$
- D$\infty$
- A
Cramer's rule
Δ≠0
Q1MCQCramerBy Cramer's rule, $x$ equals:- A$\dfrac{\Delta_1}{\Delta}$
- B$\dfrac{\Delta}{\Delta_1}$
- C$\Delta_1\Delta$
- D$\Delta-\Delta_1$
- A
Unique solution
Q1MCQUnique solutionA system has a unique solution iff:- A$\Delta\neq0$
- B$\Delta=0$
- C$\Delta_1=0$
- D$\Delta=\Delta_1$
- A
Infinite/consistent
Q1MCQInconsistentIf $\Delta=0$ but some $\Delta_i\neq0$, the system is:- Ainconsistent (no solution)
- Buniquely solvable
- Chomogeneous
- Dconsistent with infinitely many
- A
Inconsistent
Q1MCQHomogeneous$AX=0$ has a non-trivial solution iff:- A$\Delta=0$
- B$\Delta\neq0$
- C$A=I$
- D$\Delta=1$
- A
Homogeneous non-trivial
Q1NumericalIdentityThe value of $|I_3|$ is:Determinant of identity
Q1NumericalCross cofactor sum$a_{11}A_{21}+a_{12}A_{22}+a_{13}A_{23}$ (row 1 entries, row 2 cofactors) equals:Cross cofactor sum
Q1Numerical|adj A|For a $3\times3$ matrix with $|A|=2$, $|\operatorname{adj}A|$ is:Determinant of adjoint
Q1Numerical|A^{-1}|If $|A|=5$, then $|A^{-1}|$ is (as a decimal):Determinant of inverse
Q1MCQVandermonde$\begin{vmatrix}1&a&a^{2}\\1&b&b^{2}\\1&c&c^{2}\end{vmatrix}$ equals:- A$(a-b)(b-c)(c-a)$
- B$(a+b)(b+c)(c+a)$
- C$abc$
- D$a^{2}+b^{2}+c^{2}$
- A
Vandermonde
Q1NumericalSkew-symmetricThe determinant of a $3\times3$ skew-symmetric matrix is:Skew-symmetric (odd order)
Q1MCQDifferentiationTo differentiate a determinant, one:- Adifferentiates one row (or column) at a time and adds
- Bdifferentiates every entry at once
- Cmultiplies by $x$
- Dtransposes it
- A
Differentiation of a determinant
Q1NumericalRow of zerosThe value of a determinant containing a full row of zeros is:Row of zeros
Q1MCQSum propertyIf a row of a determinant is $(p_i+q_i)$, the determinant:- Asplits into two determinants
- Bis zero
- Cis doubled
- Dis unchanged
- A
Sum property of a row
Q1Numerical2×2 againThe value of $\begin{vmatrix}5&0\\0&5\end{vmatrix}$ is:
More JEE Advanced Mathematics formula sheets
Every chapter sheet is typeset, free and open without a sign-in.
- 3d36 formulas
- Area Under THE Curve30 formulas
- Binomial Theorem26 formulas
- Circle34 formulas
- Complex Number22 formulas
- Compound Angle34 formulas
- Continuity26 formulas
- Definite Integration32 formulas
- Differentiability26 formulas
- Differential Equation32 formulas
- Ellipse34 formulas
- Function36 formulas
- Fundamental OF Mathematics30 formulas
- Hyperbola36 formulas
- Indefinite Integration34 formulas
- Inverse Trigonometric Function36 formulas
- Limit34 formulas
- Logarithm30 formulas
- Matrices34 formulas
- Maxima AND Minima32 formulas
- Method OF Differentiation34 formulas
- Monotonicity26 formulas
- Parabola36 formulas
- Permutation & Combination32 formulas
- Probability34 formulas
- Properties AND Solution OF Triangles20 formulas
- Quadratic Equation26 formulas
- Sequence AND Series36 formulas
- SET and Relation38 formulas
- Statistics30 formulas
- Straight Line32 formulas
- Tangent AND Normal34 formulas
- Trigonometrical Equation28 formulas
- Vectors36 formulas
Other ways to revise this chapter
Master this chapter with similar other learning materials.
Preparing students for India’s top institutes
Our students are currently into top technological and medical institutes of India.
IIT Bombay
IIT Delhi
IIT Madras
IIT Kanpur
IIT Kharagpur
IIT Roorkee
IIT Guwahati
IIT BHU Varanasi
AIIMS Delhi
NIT Tiruchirappalli
NIT Rourkela
Join QuestPix, Today!
Get notified first, with exam & curriculum updates, course & test series launch offers, motivation & success stories and free learning resources recommended by toppers.
